- Classes of Conversions
- Distinctions Among the Three Classes
- Converting Temperature Units
- Converting Temperatures
- Converting Temperature Changes
| Bette: | Oh Prof, can we talk with you a minute? |
| Prof: | Hi Bette; sure. What seems to be the problem? |
| Alf: | We're getting confused about converting temperatures. |
| Bette: | Yeah. Say, we need to convert the gas constant R from one set of units to another. |
| Prof: | Ok, you've proposed a situation; can you state the problem? |
| Alf: | Well, let's say we have R = 8.314 J/(mol K), but we need it in Btu/(lbmol °R). The problem is, how do we convert the temperature in R? |
| Prof: | What temperature? There is no temperature in the gas constant: the Kelvin or Rankine in the gas constant is a temperature unit, not a temperature. |
| Alf: | I don't get it. |
| Prof: | Ok, let's leave temperature aside for a few minutes and begin by reviewing what you know about conversion factors. How long have you been doing unit conversions? |
| Bette: | Gee, Prof, it seems like forever; I mean, we're always doing it. |
| Prof: | So, you must have organized your thinking about them, right? |
| Alf: | Organized? |
| Prof: | A purpose of education, Alf, is to organize your thinking into economical and useful forms. |
| Bette: | What's to organize about conversion factors? You just do them. |
| Prof: | But you do seem to have a difficulty distinguishing temperatures from temperature units, so something in your thinking seems lacking. |
| Alf: | Well … |
| Prof: | Indeed. Ok, consider this: one way to organize is by classification. Can we organize conversions into classes? If we look at the conversions we routinely do, they divide into three classes. |
First are those in which we merely rename the unit. For example,
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| These are called isomorphic transformations; in such conversions, nothing changes numerically, we only change the name. The conversion factor is unity. | |
| Bette: | Why do we bother to change a name? |
| Prof: | Usually, it's a matter of economy; I'd rather write one symbol (1 N) instead of half a dozen (1 kg m/s2). Sometimes there are political motivations, such as the use of Celsius instead of the older term centigrade. |
| Alf: | Ok, Prof, what other classes are there? |
| Prof: | You tell me, Alf. |
| Alf: | Well, there are those in which the name changes and in which the conversion factor is not unity. |
| Prof: | Good. These are called similarity transformations. For us, similarity simply means a scaling; we scale one unit by a factor to get another unit. Most conversions are like this. Can you give some examples? |
| Bette: | 1 foot = 12 inches; 1 minute = 60 seconds; 1 meter = 100 cm. |
| Prof: | Good. Now, Alf, can you identify the third class? |
| Alf: | Well, we haven't gotten to temperature conversions yet. |
| Prof: | Right. The third class is composed of linear transformations; these involve a shift in the reference point (origin), with or without a scaling. For example, conversions from Celsius to Fahrenheit involve both a shift in origin and a scaling: |
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T[°F] = (9/5) T[°C] + 32
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| But conversions from Celsius to Kelvin only involve a shift in the reference point: | |
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T[K] = T[°C] + 273.15
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| Alf: | Are there other linear transformations, besides those in temperature? |
| Prof: | Bette? |
| Prof: | Wait a minute … oh, there's the conversion from gage pressure to absolute, |
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P[absolute] = P[gage] + P[atmosphere]
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| Prof: | Good. |
| Alf: | Ok, Prof, we've got three kinds of conversions: isomorphic transformations, similarity transformations, and linear transformations. So what? |
| Prof: | So, let's consider what's really happening in each of these. To begin, consider the similarity transform; we've said this is always a scaling, but a scaling of what? |
| Alf: | I'm not sure … |
| Prof: | Look at one of Bette's examples: 1 foot = 12 inches. What is being scaled when we change a measured length from feet to inches? Does the length change? |
| Bette: | Of course not, Prof. |
| Prof: | Then what? |
| Bette: | The units. |
| Prof: | And, what about the unit changes from 1 to 12? |
| Alf: | Ah! The size of the unit. |
| Prof: | Exactly. Similarity transforms scale the size of the unit; and when we change the size of the unit we also give the new size a different name. In contrast, in an isomorphic transformation, the size of the unit remains the same. |
| Alf: | But in a linear transform we change the zero for the scale of unit and we may or may not also change the size of the unit? |
| Prof: | You've got it. |
| Bette: | Ok, Prof. But how does this help us with the conversion of R problem? |
| Prof: | Remember my remark that R contains temperature units, not temperatures? |
| Alf: | So? |
| Prof: | So, converting a temperature unit is never a linear transformation; it may be isomorphic or similarity. For example,
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| And since converting among values of the gas constant R involves converting among temperature units, only scalings of those units are required. So your conversion of R requires only similarity transforms: | |
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| Bette: | Ok, but what about converting temperatures? | ||||||
| Prof: | I'm afraid we have so many possible temperature scales and units, that any kind of conversion can arise. A temperature conversion
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| Alf: | But can we make a general rule? | ||||||
| Prof: | Sure Alf, here it is: Whenever you convert any derived quantity that contains a temperature unit, the conversion requires only a scaling of the temperature unit—never a linear conversion. You merely scale by a factor; it might be that the factor is unity (isomorphic) or not unity (similarity). | ||||||
| Bette: | So our organization of conversion factors is this: | ||||||
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| Prof: | Exactly! |
| Alf: | Ok, Prof, what about converting temperature changes? |
| Prof: | Good question, Alf. A change is a difference, for example (recall that Δ is the "change" operator), |
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ΔT = T2 - T1
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| And the subtraction removes the reference point from the change. So converting a temperature change is never a linear transform; it may be similarity or isomorphic. It is similarity if the sizes of the new and old temperature units differ, but it is isomorphic if the sizes are the same. | |
| Bette: | What about some examples, Prof? |
| Prof: | Ok, Bette. Since Celsius and Kelvin have the same size unit, conversions of changes between these two are isomorphic: |
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ΔT[Kelvin] = 1 ΔT[Celsius]
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| But the sizes of Celsius and Fahrenheit units differ, so conversions of changes between these two are similarity transforms: | |
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ΔT[°F] = 1.8 ΔT[°C]
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| Alf: | Then conversions of changes between Fahrenheit and Rankine must also be isomorphic? |
| Prof: | That's right, Alf. |
| Bette: | Great Prof. Thanks a lot. |
| Prof: | No problem. |
